Does gradient descent in deep learning assume a smooth fitness landscape?
I'm going to take the fitness landscape to be the graph of the loss function, $\mathcal{G} = \{\left(\theta, L(\theta)\right) : \theta \in \mathbb{R}^n\}$, where $\theta$ parameterises the network (i.e. it is the weights and biases) and $L$ is a given loss function; in other words, the surface you would get by plotting the loss function against its parameters.
We always assume the loss function is differentiable in order to do backpropagation, which means at the very least the loss function is smooth enough to be continuous, but in principle it may not be infinitely differentiable1.
You talk about using gradient descent to find the global minimiser. In general this is not possible: many functions have local minimisers which are not global minimisers. For an example, you could plot $y = x^2 \sin(1/x^2)$: of course the situation is similar, if harder to visualise, in higher dimensions. A certain class of functions known as convex functions satisfy the property that any local minimiser is a global minimiser. Unfortunately, the loss function of a neural network is rarely convex.
For some interesting pictures, see Visualizing the Loss Landscape of Neural Nets by Li et al.

1 For a more detailed discussion on continuity and differentiability, any good text on mathematical analysis will do, for example Rudin's Principles of Mathematical Analysis. In general, any function $f$ that is differentiable on some interval is also continuous, but it need not be twice differentiable, i.e. $f''$ need not exist.